Question 1200771
**1. Determine the Minimum Number of Grids for a Single Droplet**

* **Probability of Absorption by a Single Grid:**
    * The droplet will be absorbed by a single grid if its center falls within a circle of radius r + 0.5 (since the grid size is 1x1 and the droplet has radius r).
    * Area of the circle: π * (r + 0.5)² 
    * Area of the grid: 1 * 1 = 1
    * Probability of absorption by a single grid: (π * (r + 0.5)²) / 1 = π * (r + 0.5)²

* **Probability of Absorption by 'n' Grids (Cumulative Probability):**
    * Assuming independent events, the probability of the droplet not being absorbed by any of the 'n' grids is:
        * (1 - Probability of absorption by a single grid)^n 
        * = (1 - π * (r + 0.5)²) ^ n

    * Therefore, the probability of the droplet being absorbed by at least one of the 'n' grids is:
        * 1 - (1 - π * (r + 0.5)²) ^ n

* **Find 'n' for Desired Probability (α):**
    * 1 - (1 - π * (r + 0.5)²) ^ n ≥ α
    * (1 - π * (r + 0.5)²) ^ n ≤ 1 - α
    * n * log(1 - π * (r + 0.5)²) ≤ log(1 - α) 
    * n ≥ log(1 - α) / log(1 - π * (r + 0.5)²)

* **Substitute given values:**
    * α = 0.72, r = 0.1
    * n ≥ log(1 - 0.72) / log(1 - π * (0.1 + 0.5)²) 
    * n ≥ log(0.28) / log(1 - π * 0.36) 
    * n ≥ 2.04 

* **Since 'n' must be an integer, the minimum number of grids is 3.**

**2. Calculate Fraction of Uncovered Grids for the Stream**

* **Probability of a droplet of radius 'r' being absorbed:** 0.72 (given)
* **Probability of a droplet of radius 's' being absorbed:**
    * Calculate using the same formula as in step 1:
        * 1 - (1 - π * (s + 0.5)²) ^ 3 
        * 1 - (1 - π * (0.13 + 0.5)²) ^ 3 
        * ≈ 0.836

* **Probability of a droplet NOT being absorbed:**
    * Droplet of radius 'r': 1 - 0.72 = 0.28
    * Droplet of radius 's': 1 - 0.836 = 0.164

* **Fraction of uncovered grids:**
    * (0.21 * 0.28) + (0.79 * 0.164) ≈ 0.217

**Therefore:**

* **Minimum number of grids:** 3
* **Fraction of uncovered grids:** 0.217

**Note:**

* This calculation assumes that the droplets fall independently and uniformly on the grid surface.
* This analysis provides an estimate. Actual results may vary depending on the specific distribution of droplets within the stream.